Notes on Units and Physical Quantities

The Nature of Physics

• Physics is an experimental science in which physicists seek patterns in several natural phenomena.
• The patterns are called physical theories.
• A very well established or widely used theory is called a physical law or principle.
• This course introduces how to describe and analyze physical phenomena using quantities, units, and measurements.

Physical Quantity

• A physical quantity is any number that is used to describe a physical phenomenon.
• Examples of physical quantities:
  • Force: $F = 30\;\text{N}$
  • Time: $60\;\text{s}$
  • Length: $1.0\;\text{m}$
  • Mass: $50\;\text{kg}$
• The term “standard magnitude” is used to illustrate a representative numerical value for a quantity.

Fundamental Units (SI Base Units)

• The International System (SI base units) is also known as the metric system.
• Seven base quantities and their units:
  • Length: name = \text{meter}, symbol = $\text{m}$
  • Mass: name = \text{kilogram}, symbol = $\text{kg}$
  • Time: name = \text{second}, symbol = $\text{s}$
  • Electric current: name = \text{ampere}, symbol = $\text{A}$
  • Thermodynamic temperature: name = \text{kelvin}, symbol = $\text{K}$
  • Amount of substance: name = \text{mole}, symbol = $\text{mol}$
  • Luminous intensity: name = \text{candela}, symbol = $\text{cd}$
• These base units form the foundation from which all other (derived) units are constructed.

Dimensional Consistency

• A physical equation must be dimensionally consistent: terms that are added or equated must have the same units.
• Always compare similar quantities (apples to apples).
• Example of an incorrect dimensional match: $\text{Mass} \neq \text{Length}$
• Dimensional analysis helps verify the consistency of equations and the dimensions of quantities such as velocity, acceleration, etc.
• For example, if velocity is expressed as a product of acceleration and time, then the dimensions must satisfy:
  • Velocity = acceleration × time
  • $[v] = [a][t] = \left(\frac{\text{m}}{\text{s}^2}\right)(\text{s}) = \frac{\text{m}}{\text{s}}$
  • Hence, $[v] = [L][T]^{-1}$ and $[a] = [L][T]^{-2}$.

Dimensional Analysis

• Uses of dimensional analysis:
  • Derive an equation.
  • Check if an equation is dimensionally correct.
  • Determine the units or the dimension of a physical quantity.
  • Check/simplify the dimension of the LHS and RHS of an equation.

Dimensionally Correct Equations (Examples to Check)

• Problem: Determine if the following are dimensionally correct:
  1) $x = vt$
  2) $v = m + 2ax$
  3) $x = vt + \frac{1}{2}at^2$
• Given symbols:
  • x has dimension $[L]$
  • m has dimension $[M]$
  • v has dimension $\left[\frac{L}{T}\right]$
  • t has dimension $[T]$
  • a has dimension $\left[\frac{L}{T^2}\right]$

Dimensionally Correctness Check (Worked)

• 1) $x = vt$
  • LHS: $[x] = [L]$
  • RHS: $[v][t] = \left[\frac{L}{T}\right][T] = [L]$
  • Therefore, dimensionally correct.
• 2) $v = m + 2ax$
  • LHS: $[v] = [\frac{L}{T}]$
  • RHS: $[m] + 2[a][x] = [M] + \left[\frac{L}{T^2}\right][L] = [M] + [L^2 T^{-2}]$
  • Mismatch: cannot add [M] to [L^2 T^{-2}]; dimensionally incorrect.
• 3) $x = vt + \frac{1}{2}at^2$
  • RHS terms: $[v][t] = [L],\; \frac{1}{2}[a][t]^2 = \frac{1}{2}\left[\frac{L}{T^2}\right][T^2] = [L]$
  • Sum of two length terms; LHS: $[x] = [L]$; dimensionally correct if the sum of two L quantities is meaningful.

Conversion of Units

• Example: 18 years old ⇒ how many seconds?
• Given: 18 years ≈ 568,036,800 seconds; exactly $5.680368 \times 10^{8} \text{ s}$
• Conversion path (typical):
  • 1 year ≈ 365.25 days
  • 1 day = 24 hours; 1 hour = 3600 seconds
• Expression: $18\ \text{years} \times \frac{365.25\ \text{days}}{\text{year}} \times \frac{24\ \text{h}}{\text{day}} \times \frac{3600\ \text{s}}{\text{h}} = 5.680368 \times 10^{8} \text{ s}$

Significant Figures & Scientific Notation

• Final answers should be expressed with the number of significant figures corresponding to the given quantities.
• Example: 5 s.f. (as a guideline in the course).
• Example conversions:
  • $1.0 \times 10^{4} \text{ m}$ has 2 significant figures.
  • $1.04 \times 10^{4} \text{ m}$ has 3 significant figures.
  • $10\,351\ \text{m}$ has 5 significant figures.

Uncertainty, Accuracy, and Precision

• Scenarios illustrating uncertainty vs. measurement practice:
  • A cardboard thickness measured with a ruler is recorded as $3\text{ mm}$, not $3.00\text{ mm}$ (uncertainty not shown as extra decimals).
  • A vendor does not quote a precise weight like $2.41735\text{ kg}$; instead, a reasonable display of uncertainty/precision is used.
• Distinctions:
  • Accuracy: closeness to the true value.
  • Precision: reproducibility or consistency among repeated measurements.
• Visual intuition often shown as a 2x2 grid: Low/High accuracy vs Low/High precision (examples omitted here; concept explained).

Accuracies / Uncertainties in Notation

• Common representations of measurement uncertainty:
  • $56.47 \pm 0.02\ \text{ mm}$
  • $56.47(2)\ \text{ mm}$ (the (2) indicates the uncertainty in the last digits; here ±0.02 mm)
  • $47\,\Omega \pm 10\%$

Significant Figures: Rules and Examples

• When uncertainties are not specified, use the number of meaningful digits (significant figures).
• Example rules:
  • $2.91\text{ mm} \Rightarrow 3\text{ sig figs}$
  • The last digit is in the hundredths place; uncertainty about $0.01\text{ mm}$.
  • $137\text{ km} \Rightarrow 3\text{ sig figs}$; uncertainty about $\sim 1\text{ km}$.

Rules for Significant Figures

• All non-zero digits are significant.
• Zeros before or after a decimal point are significant only if preceded by a non-zero digit.
• If there is no decimal point, zeros to the right of the rightmost non-zero digit are not significant.
• Examples:
  • $1200 \quad\text{(2 sig figs)}$
  • $1200.0 \quad\text{(5 sig figs)}$
  • $13.20 \quad\text{(4 sig figs)}$
  • $112000 \quad\text{(3 sig figs)}$
  • $112000. \quad\text{(6 sig figs)}$
  • $0.0034560 \quad\text{(5 sig figs)}$

How Many Significant Figures?

• Determine sig figs for each quantity:
  • $12 \Rightarrow 2\ \text{SF}$
  • $12.0 \Rightarrow 3\ \text{SF}$
  • $0.012 \Rightarrow 2\ \text{SF}$
  • $0.0120 \Rightarrow 3\ \text{SF}$
  • $120. \Rightarrow 3\ \text{SF}$
  • $1.2 \times 10^{2} \Rightarrow 2\ \text{SF}$

Significant Figures in Operations

• Multiplication/Division: the result should have the same number of significant figures as the factor with the fewest significant figures.
  • Example: $\frac{(0.745)(2.2)}{3.885} = 0.424… \Rightarrow 0.42$ (2 SF)
• Addition/Subtraction: the result should have the same number of decimal places as the quantity with the fewest digits to the right of the decimal point.
  • Example: $27.153 + 138.2 - 11.74 = 153.613 \Rightarrow 153.6$ (1 decimal place)

Rounding Rules

• General rules for rounding:
  • If the digit X to the right is 6–9, round up; if 0–4, round down.
  • If X = 5 with nonzero digits to the right, round up.
  • If X = 5 and there are no nonzero digits to the right, round to the nearest even digit to the left (banker's rounding):
  • Example: $2.15 \to 2.2$
  • Example: $6.2500 \to 6.2$

Significant Figures in Operations (Summary)

• Multiplication/Division: fewest SF among factors.
• Addition/Subtraction: fewest decimal places among operands.

Credits/References

• University Physics 13th Ed, H. Young and R. Freedman, Pearson Education 2014
• PowerPoint Lectures for University Physics 13th Ed, Wayne Anderson, Pearson Education 2012
• Physics 71 Lectures by J Vance, A Lacaba, P J Blancas, G Pedemonte
• Annotations by: Mark Ivan Ugalino
• Edited by: Rene L. Principe Jr.